# The hypergroupoid of boundary conditions for local quantum observables

December 09, 2016

We review the definition of hypergroups by Sunder, and we associate a
hypergroup to a type III subfactor $N\subset M$ of finite index, whose
canonical endomorphism $\gamma\in\mathrm{End}(M)$ is multiplicity-free. It is
realized by positive maps of $M$ that have $N$ as fixed points. If the depth is
$>2$, this hypergroup is different from the hypergroup associated with the
fusion algebra of $M$-$M$ bimodules that was Sunder's original motivation to
introduce hypergroups.
We explain how the present hypergroup, associated with a suitable subfactor,
controls the composition of transparent boundary conditions between two
isomorphic quantum field theories, and that this generalizes to a hypergroupoid
of boundary conditions between different quantum field theories sharing a
common subtheory.

Keywords:

algebraic quantum field theory, hypergroups, defect, boundary conditions